Lederman Roy R, Steinerberger Stefan
Department of Statistics and Data Science, Yale University, New Haven, CT 06511, USA.
Department of Mathematics, University of Washington, Seattle, WA 98195, USA.
Linear Algebra Appl. 2024 Dec 15;703:528-555. doi: 10.1016/j.laa.2024.09.014. Epub 2024 Sep 27.
Let be a tree on vertices and let denote the Laplacian matrix on . The second-smallest eigenvalue , also known as the algebraic connectivity, as well as the associated eigenvector have been of substantial interest. We investigate the question of when the maxima and minima of an associated eigenvector are assumed at the endpoints of the longest path in . Our results also apply to more general graphs that 'behave globally' like a tree but can exhibit more complicated local structure. The crucial new ingredient is a reproducing formula for eigenvectors of graphs.
设(T)是一个具有(n)个顶点的树，并用(L)表示(T)上的拉普拉斯矩阵。第二小特征值(\lambda_2)，也称为代数连通性，以及相关的特征向量一直备受关注。我们研究与特征向量相关的最大值和最小值何时在(T)中最长路径的端点处取得的问题。我们的结果也适用于更一般的图，这些图在全局上“表现得像树”，但可能具有更复杂的局部结构。关键的新要素是图特征向量的一个再生公式。
